C₇×#9020 #13186236

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#9020

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂×C₇×#392	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

1224 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5,6,7)(9,11)(12,13,17,18,16,14,15,19)(20,21)(22,26,27,24)(23,25);(8,9,10,11)(12,14,16,13)(15,19,17,18)(20,21)(22,24)(23,25)(26,27);(12,15,16,17)(13,18,14,19)(20,22,23,27)(21,24,25,26)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2008);